To answer my own question, here is what happened:

I prepaid the expected charge, and then I payed the outstanding balance (plus a little extra for expected charges). Then, 1 week ahead, I called Amex and advised them of the upcoming charge. They told me that they do not advise prepaying, but, in essence, that they can't stop me from doing it (it would have been too late anyway). They asked me how much the upcoming charge was and at what business and if it was over the phone (no, it was in person, card present). The agent said she put a note on my account, but that the charge may still be declined.

A week later, when I tried to pay, the charge was in fact declined. The clerk at the business where I tried to pay told me that the error message was an unusual one she had never seen before. I called Amex. They told me that the business had to call Amex themselves, and I was given a certain phone number for that.

The clerk of the business called the number. She was asked several questions, then the charge was put through again - and this time it was accepted.

However, my Amex credit card was rendered useless thereafter. Although I still had a small positive balance, the card was being refused for other charges. Again, I called Amex and was told that I couldn't even charge a single dollar to the card until the large charge had been cleared, which was to take a number of days.